In this exercise, you will analyze building height variation in a city where buildings of different ages coexist. As in the example in chapter 3, suppose that the city grows outward by one block each year. Block 0 is built in year 0, block 1 is built in year 1, block 2 is built in year 2, and so on. Suppose also that buildings are torn down and replaced after standing for 4 years. Buildings built in year 0 are replaced in year 4, buildings built in year 1 are replaced in year 5, and so on.

(a) Derive the city's building-age pattern in year 11, when it has a radius of 12 blocks (0-11). Once the age pattern is derived, write down the year in which buildings in each block were built.

(b) Suppose that the height of buildings depends on their location and their construction date. For a given construction date, buildings farther from the CBD are shorter. At a given location, buildings built later in time are taller. Let S equal the height of a building measured in stories, and suppose that S = 5T - 2x, where T is the construction date (the number of the year in which the building was built) and x is the number of the block in which the building is located. Using this formula and the results of (a), compute building heights in each block of the city in year 11. Plot your results.

(c) Now suppose that building heights are determined by a different formula: S = 5T - 5x + 10. Repeat (b) for this case.

(d) Contrast the patterns shown in your plots to the building-height pattern predicted by the model from chapter 2, where building height is continually adjusted in response to changing conditions.